Incoherent nonadiabatic to coherent adiabatic transition of electron transfer in colloidal quantum dot molecules

Electron transfer is a fundamental process in chemistry, biology, and physics. One of the most intriguing questions concerns the realization of the transitions between nonadiabatic and adiabatic regimes of electron transfer. Using colloidal quantum dot molecules, we computationally demonstrate how the hybridization energy (electronic coupling) can be tuned by changing the neck dimensions and/or the quantum dot sizes. This provides a handle to tune the electron transfer from the incoherent nonadiabatic regime to the coherent adiabatic regime in a single system. We develop an atomistic model to account for several states and couplings to the lattice vibrations and utilize the mean-field mixed quantum-classical method to describe the charge transfer dynamics. Here, we show that charge transfer rates increase by several orders of magnitude as the system is driven to the coherent, adiabatic limit, even at elevated temperatures, and delineate the inter-dot and torsional acoustic modes that couple most strongly to the charge transfer dynamics.

quantum dot dimmer with thousands of atoms by combining Mixed quantum-classical Ehrenfest dynamics and a semi-empirical pseudopotential model. The highlight quantum dot dimmer behaves from the adiabatic to the nonadiabatic limit of electron transfer by turning the neck and core sizes and decreasing the shell thickness.
The topic of charge transfer in this article is interesting. Using a series of Quantum Dot Molecules to probe different electron transfer regimes is an exciting challenge. However, I have a few general issues with the manuscript in its current form that prevent me from recommending it for publication.
1. Zhu et.al have reported a relization of the adiabatic and nonadiabatic electron transfer by changing the bridge length connecting the donor and acceptor and the functional groups in 2021. The manuscript has limited novelty. 2. In figure 2c, the authors found an excellent correlation between J/lamda and the adiabatic parameter. J is the coupling energy between the doner and acceptor states, and lamda is the reorganization energy. With the physical picture of adiabatic and diabatic potential energy surfaces in figure 1, the reasonable range of J/lamda should be smaller than 0.5. Please clarify the range of J/lamda. Response to Reviewer 1.

Comment 1.
In this work, Rabani and coworkers report on a theoretical framework to analyze different regimes of charge transfer between two colloidal quantum dots connected by a bridge. By varying the characteristic parameters of the quantum dot building blocks -neck size, core size, or shell thickness -the authors show that their atomistic model can reproduce different regimes of electron transfer, from Marcus nonadiabatic transfer to the adiabatic limit.
This work presents an impressive set of charge-transfer atomistic simulations of colloidal quantum dots that unravel the passage through different electron-transfer regimes. While I believe that this work could be of interest to the readership of Nature Communications, I have a few concerns that the authors may need to address.

Reply 1.
We thank the reviewer for the positive assessment of our work and for finding it interesting for the readership of Nature Communications.

Comment 2.
My first concern is linked to the lack of validation of the theoretical formalism to reproduce experimental observables related to the colloidal quantum dots. If the goal of this work is to stimulate new experiments on this system, I would have perhaps expected that the authors find some observables to benchmark and validate their theoretical framework, bringing more confidence in the predictions discussed here. Ref. [12] reports, for example, absorption spectra. Would it be possible for the authors to provide further validation of their theoretical formalism?

Reply 2.
We thank the reviewer for the comment. The reviewer concerns about the validation of our theoretical formalism related to the experimental observables in colloidal quantum dots. In fact, we have validated our theoretical models on both electronic, exitonic and vibronic properties of the colloidal quantum dot monomers and dimers in previous work.
For examples: (1) Using the semiempirical pseudopotential method combined with the Bethe-Salpeter equation (BSE) [1] in the static screening approximation [2] we have calculated the optical gap and compared our predictions with experimental measurements for several different NC compositions and size, as illustrated in Fig. 1. We find very good agreement between the computed and measured gaps across all systems considered, validating our pseudopotential model and the approach.  [3]. Gaps for wurtzite CdSe quantum dots of different sizes (left). The optical gaps computed by our semiempirical pseudopotential method agree with experimental measurements of the optical gap by Fan et al. [4] (black squares) and Yu et al. [5] (black triangles). The inset shows the exciton binding energy, E B , computed by our method and compared to the values computed by Franceschetti and Zunger [6] (black asterisks). Gaps for zincblende CdSe-CdS core-shell nanoplatelets with different thicknesses of the CdS shell (middle). The optical gaps calculated by our method compare favorably with those measured experimentally by Hazarika et al. [7] (black squares). Gaps for zincblende InAs quantum dots of different sizes (right). The fundamental gaps calculated are in excellent agreement with those measured by Banin et al. [8] using scanning tunneling microscopy (black squares), and the optical gaps compare well with those measured by Guzelian et al. [9] (black triangles) and computed by Franceschetti and Zunger [6] (black asterisks).
(2) The reorganization energies (correlate with the experimentally measured Stokes shifts) for wurtzite CdSe calculated using exciton-phonon coupling obtained from the pseudopotential model are shown in Fig. 2). We find that with no adjustable parameters, the cal-    [10]. Reorganization energies of exciton for CdSe NCs of various diameters in comparison to values from effective mass model-based calculations by Kelley [11] and from experimental measurements by Bawendi et al. [12] and Scholes et al. [13] (left). Acoustic modes contribute more significantly to the reorganization energy than optical modes (right). order coupling models), and compared to single NC PL measurements (green curves). Our calculations are in very good agreement with the single-NC PL measurements at low and intermediate temperatures, providing a quantitatively accurate description of the relative positions and intensities of the zero-phonon line (ZPL) and the phonon sidebands, as well as their temperature dependence. At higher temperatures, the inclusion of the second-order exciton-phonon coupling is required to account for high temperature lattice fluctuations.
(4) We have also calculated the nonradiative Auger recombination (AR) rates using the pseudopotential model with the BSE and Fermi's golden rule, and compared our prediction  [14]. Single-molecule photoluminescence spectra for a 3nm diameter CdSe / 3 monolayer CdS core-shell nanocrystal at temperatures ranging from 4 to 290 K. The calculated results from the model Hamiltonian (black curves) and from the empirical inclusion of second-order expansion of exciton-phonon couplings (red curves) are compared to the experimental measurements (green curves).  [15]. AR lifetimes, τ AR , for CdSe QDs as a function of the volume of the QD. Good agreement is observed between the interacting formalism (green circles) and experimental (blue squares: solid [16], vertical lines [17], and horizontal lines [18]) AR lifetimes for all sizes. On the other hand, the noninteracting formalism (red triangles) deviates from the experimental values for QD volumes >10 nm 3 . Power law fits, τ AR = a × V b , are also shown for each of the three sets of AR lifetimes.
with experimental measurements of the AR lifetimes, [16][17][18] as shown in Fig. 4. The overall agreement with the experiments is remarkable as well as the first approach to capture the "universal" volume scaling of AR lifetimes.
(5) In Fig. 5(a) we compare the optical shift between the QD monomers and the corresponding QD dimers, using the semiempirical pseudopotential method combined with the BSE.
We find an excellent agreement between the measured (red stars) and the calculated (green squares) shifts when comparing similar shell thicknesses (R S = 2.1nm). Our model allows delineating the role of strain, deconfinement, and hybridization. In summary, previous validations of our approach indicate that it is reliable in predicting the exciton optical gap, reorganization energy/Stokes shift, photoluminescence spectral line shape, Auger recombination rate, and exciton hybridization energies in QD dimer. In order to prevent confusion, we have added citations to the above validations, and added the following sentence on page 3, "The approach was recently validated in predicting optical gap, reorganization energies/Stokes shifts, linear optical PL spectrum, multi-excitonic effects, and more [14][15][16][17][18]."

Comment 3.
Connected to this first issue, the authors use Ehrenfest dynamics for the three studied regimes. Could the authors possibly provide a validation for using this mixed quantum/classical method for the different types of charge-transfer mechanisms under investigation? Any shortcomings emerging from its mean-field nature?

Reply 3.
We thank the reviewer for the comment. The performance of Ehrenfest dynamics can be validated by comparing it with numerically exact methods such as the multi-configurational time-dependent Hartree (MCTDH) approach [21][22][23][24]. The MCTDH calculations done here for validation were performed with the Heidelberg MCTDH package [25]. Fig. 6 shows the comparison of population dynamics calculated from Ehrenfest and MCTDH approach in  For the step of integration, we use ∆t = 1 fs for integrating both system and phonon dynamics. We use 4th order Runge-Kutta methods to integrate the equation of motion, whose local truncation error scales as O(∆t 5 ) (convergence plot of ∆t is shown in Fig 7).
Note that the bath equation of motion is integrated numerically taking ρ S (t) to be constant over a half-time step We rewrite our description in page 10 as "The above equations of motion in (12)(13)(14) can be integrated by the fourth order Runge-Kutta method with a time step around 1 fs. The phonon coordinates are integrated taking ρ S (t) to be constant over a half-time step". The manuscript is well written and is well prepared. The results seem to be sound (as far as one can check this without repeating the simulations).

Reply 1.
We thank the reviewer for the in-depth review and support for our results. In the following, we will address the comments one by one.

Comment 2.
My main problem with the manuscript is that I did not get excited about the results at all. My background is not in electronic transfer, but I work on quantum dots and optical passages, for which I see a lot of similarities to what is found in this study. Hence, I was able to follow the manuscript easily. But I could neither see the big novelty in the manuscript.

Reply 2.
We thank the reviewer for the valuable comments and for being able to follow our manuscript, despite being outside the community. Electron transfer is indeed an "old" problem. Most of our intuition and understanding of electron transfer relies on intra and intermolecular electron transfer in solution on one hand and molecular junctions, molecule/semiconductor, and semiconductor/semiconductor interfaces on the other hand.
Such control is required in order to drive the mechanism of electron transfer and achieve ultrafast charge transfer devices. Indeed, only very recently it was demonstrated that such a transition can be systematically tunned in a multi-valence donor-bridge-acceptor complex [26]. In comparison, our system offers more controls over the donor and acceptor couplings, in a regime that results in ultrafast coherent electron transfer at room tempera-tures, rather than the over-damped molecular limit [26]. Coherence at room temperature is also important for other quantum information-based applications. We believe that while our work is computational and predictive, our model is well tested (see reply to Referee 1), and the novel results will generate interest in the community in both confirming our predictions as well as in searching for optimal charge transfer devices based on quantum dot dimers. We also added "The flexibility of controlling the hybridization energy without affecting the other energy scales in the system, offers a platform to drive the system from the over-damped electron transfer dynamics typical to molecular systems to the coherent limit, where the electron transfer rates are governed by decoherence times, even at elevated temperatures." on page 3 to emphasize our novelty.

Comment 3.
The paper strongly uses the community language, doesn't explain what is meant by adiabatic/non-adiabatic or what the larger picture is.

Reply 3.
We thank the reviewer for the comment. We have better explained what is meant by nonadiabatic in the main text by modifying the introduction into "The theoretical framework to describe charge transfer reactions in condensed phases dates back to the seminal work of Marcus [1], where he considered a donor molecule weakly coupled to an acceptor molecule, and developed a theoretical framework to describe the electron transfer reactions in a fluctuating environment." and "This nonadiabatic limit is characterized by a small value of the so-called "adiabatic parameter", γ, defined as the ratio between the donor-acceptor hybridization (J), the characteristic nuclear vibrational (ω c ), and the reorganization energy λ (see Fig. 1(a) for a sketch of these energy parameters) [2]: The nonadiabatic electron transfer Marcus regime is thus achieved either for weak electronic coupling and/or for fast nuclear motion (ω c > J/h).", when we discuss Fig.1, on page 2.
We also add "Zusman developed a framework to describe the crossover from the Marcus weak coupling nonadiabatic limit to the adiabatic limit [7,8], where the coupling between donor and acceptor is large and/or the nuclei are slow (γ ≫ 1). In the adiabatic limit, diabatic states are no longer a good representation and the dynamics proceed on a single adiabatic (Born-Oppenheimer) surface (see solid lines in Fig. 1(a))." when discussing adiabatic electron transfer on page 3.

Comment 4.
Therefore, I do not think that this paper is a good fit to Nature Communication, but could be well published in a more specialized journal.

Reply 4.
We thank the reviewer for providing valuable comments and suggestions, which helped us improve the manuscript. We hope that our reply to the referee's comments and to the comments of the other referees changes the reviewer's mind about our work. We feel, as do the other two referees, that the work (with proper adjustments) does present work suitable for publication in Nature Communications.
Response to Reviewer 3. The topic of charge transfer in this article is interesting. Using a series of Quantum Dot Molecules to probe different electron transfer regimes is an exciting challenge. However, I have a few general issues with the manuscript in its current form that prevent me from recommending it for publication.

Reply 1.
We thank the reviewer for the in-depth review and high evaluation of our work. In the following, we will address the comments one by one. transitions from nonadiabatic to adiabatic and incoherent to coherent electron transfer can be realized. Second, we find that the QD dimers have better tunability over molecular systems since changing the electronic couplings can be achieved by increasing the neck sizes or core-to-shell ratio, and does not significantly affect the reorganization and characteristic vibrational frequency. Therefore, we believe our study provide a novel platform to realize the diverse behavior of electron transfer.
Comment 3. figure 2c, the authors found an excellent correlation between J/lamda and the adiabatic parameter. J is the coupling energy between the doner and acceptor states, and lamda is the reorganization energy. With the physical picture of adiabatic and diabatic potential energy surfaces in figure 1, the reasonable range of J/lamda should be smaller than 0.5. Please clarify the range of J/lamda.

Reply 3.
We thank the reviewer for the comment. The potential energy surface in Fig.1 (a)  regime. In this case, there will be no energy barrier and the dynamics will be coherent.
To avoid possible confusion, we add "(a) Schematic sketch of the potential energy surfaces along the reaction coordinate [1] Q for the nonadiabatic electron transfer and the energy scales appearing in Eq. (1)." in the caption of Fig. 1(a).  3. Please clarify the physical source of the dephasing of electron transfer.

Reply 4.
We thank the reviewer for the comment. As indicated in the manuscript, there are several factors that contribute to the dephasing of electron transfer. We showed in Fig. 4(a) (in the main text) that the spectral density S(ω) peaks at around 0.6 THz, which means that the electronic system couples most strongly to those low-frequency vibrations. Thus, those lowfrequency vibrations contribute to the time scale of dephasing in the population dynamics. Fig. 9 below compares the population dynamics with and without the vibrational modes below the characteristic frequency ω c /2π = 0.6 THz, which shows that the dephasing time becomes much longer when removing those low-frequency modes. We modify the sentence on page 6 to emphasize this "We also find that the dephasing rate, physically resulting from coupling to the nuclear vibration, increases with increasing hybridization energy, J, deep in the adiabatic limit (γ ≫ 1).". We also added a section called "Physical Source of Dephasing" in the supporting information to discuss this point.
Another source of dephasing comes from populating different donor and acceptor states.
As shown in Fig. S2(c) in the Supporting Information, the population oscillates between D 1 /A 1 (1S e -like states) and D 2 /A 2 (1P e -like states), with different dephasing rates, controlled by the off-diagonal couplings V α n̸ =m (in main text Fig. 4(b)) to the phonons.

Reply 5.
We thank the reviewer for the valuable suggestion. Fig. 1 in the main text has been modified into Fig. 10    From my point of view, the paper is technically sound and the authors took care to use valid approaches. As previously mentioned, the paper is well written and easy to follow. Hence, the paper is surely publishable.
The main concern that I raised is that the paper has a rather limited audience and is not accessible to people outside the community. Response to Reviewer 1.

Comment
I thank the authors for their thorough and convincing answers to my questions. I would support publication of this revised manuscript in Nature Communications.

Reply
We would like to thank the reviewer for the thoughtful comments and suggestions on our original manuscript.
Response to Reviewer 2.

Comment
This is my second report to the manuscript "Nonadiabatic to Adiabatic Transition of Electron Transfer in Colloidal Quantum Dot Molecules" by Huo et al. Three reviews were performed and the authors have answered to the questions raised.
From my point of view, the paper is technically sound and the authors took care to use valid approaches. As previously mentioned, the paper is well written and easy to follow.
Hence, the paper is surely publishable.
The main concern that I raised is that the paper has a rather limited audience and is not accessible to people outside the community. The authors have replied to my comment and added a few remarks to the paper. While these comments have improved the manuscript, I am still not excited about it and still think the value of the work is not easily accessible for a general audience. But I can acknowledge that coherence at room temperature is a very crucial task for quantum technologies and that the research might spark experiments.
Thus, it might fit into the scope of Nature Communications.

Reply
We would like to thank the reviewer for the thoughtful concerns and positive evaluation on our paper. We believe that our revised version is more accessible for the general audience and hope that our predictions would lead to further experimental work.
In the revised version of "Incoherent Nonadiabatic to Coherent Adiabatic Transition of Electron Transfer in Colloidal Quantum Dot Molecules", the authors provided further explanations to confirm the novelty. And I am satisfied with the author's response to my comments. And recommend the current version could be published in NATURE COMM without further revisions.

Reply
We would like to thank the reviewer for the thoughtful comments and suggestions on our original manuscript.